Stat 501/502 Comprehensive Exam - Cherry
Comprehensive Exam
Probability and Mathematical Statistics
August 19, 2013
Note: Start each problem on a new page.
1. (5pts) Show that if A and B are independent then Ac and B c are independent.
2. (7pts) Suppose PH, a calculator company has 3 product inspectors. Let E denote the event that
an error occurred and Ii denote inspector i, i = 1, · · · , 3. The probability a defective product
slips by an inspector is
(
0.05 I1
P (E|Ii ) =
0.10 I2 and I3
Assume that inspectors I1 , I2 each inspect 40% of the products and and I3 inspects 20% of the
products. Given a defective product slips by, what is the probability that inspector 3 was the
culprit? Show all work.
3. (8pts) Let X1 , · · · , Xn and Y1 , · · · , Yn be two independent random samples each of size n from
distributions with respective means µX and µY and common variance σ 2 . Both means and the
common variance are finite. What is the limiting distribution of
i
√ h
n X n − Y n − (µX − µY )
√
.
σ 2
Be sure to justify your answer.
4. (7pts) Let X and Y be continuous random variables whose expectations exist. Show that for any
integrable function g
E (g(X)Y ) = E (g(X)) E (Y |X) .
f (x) = (θ + 1)f1 (x) − θf2 (x)
for x > 0 and 0 ≤ θ ≤ 1 and f (x) = 0 for x ≤ 0.
5. (5pts) Let X ∼ Expon(θ) and Y ∼ Expon(λ) with X and Y independent. Let Z = max(X, Y ).
Find fZ (z).
6. Let X1 and X2 be independent Expon(1) random variables.
(a) (8pts) Let U = X1 − X2 and V = X1 + X2 . Find the joint distribution of (U, V ).
(b) (5pts) Are U and V independent? Justify your answer.
7. Consider two random variables X and Y with joint pdf
(
fXY (x, y) =
(a) (5pts) Find fX|Y (x|y).
1/2; 0 < y < x < 2
0;
elsewhere
(b) (3pts) Find E(X|Y = y).
(c) (5pts) Find P (X ≤ 1|Y = y).
8. (6pts) Let X|Y ∼ N (Y, 1) and Y ∼ N (µ, σ 2 ). Find the E(X) and V ar(X).
9. Suppose X ∼ U nif orm(0, θ) with θ ∼ P areto(α, β) parameterized as
π(θ) =
βαβ
I(α,∞) (θ).
θβ+1
We consider α > 0 and β > 0 to be known constants.
(a) (5pts) Find the posterior density of θ.
(b) (3pts) Find the Bayes Estimator of θ under squared error loss.
iid
10. Suppose Xi ∼ Geom(θ), i = 1, · · · , n.
(a) (8pts) Find the MLE of θ.
(b) (15 pts) Give the large sample approximate distribution of the MLE and use it to find an
approximate 100(1 − α)% confidence interval for θ.
11. Let X1 , · · · , Xn be a random sample from the density
f (x|θ) =
2x
I(0,θ) (x)
θ2
(a) (5pts) Show that the the MLE of θ is Yn = max(X1 , · · · , Xn ).
(b) (10pts) Verify Yn is a complete sufficient statistic.
(c) (10pts) Is there a UMVUE of θ? If so, find it. If not - justify your answer.
(d) (3pts) Are the theoretical results involving the Cramer-Rao Lower Bound relevant for this
problem? Why or why not?
iid
iid
12. (10pts) Suppose X1 , · · · , Xn ∼ P oisson(θ1 ) and Y1 , · · · , Ym ∼ P oisson(θ2 ). We wish to test
H0 : θ1 = θ2 = θ0 versus Ha : θ1 6= θ2 . Find an approximate large-sample level α likelihood ratio
test. Do not spend a lot of time trying to simplify the test statistic beyond the obvious.